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Let $M$ be a differentiable manifold of dimension $n>1$ and $g$ a flat Riemannian metric on $M$. Consider $f:M\rightarrow \mathbb{R}^+$ a continuous function which doesn't have continuous first derivatives and $g':=e^{2f}g$ a non-smooth Riemannian metric on $M$.

Does the notion of curvature (curvature tensor, sectional curvature) still make sense for $(M,g')$? I'm searching for a reference which covers this aspect of singular riemannian metrics.

In the smooth case there is a formula to compute the curvature tensor $R'$ of $g'$ with respect to the derivatives of $f$:

$$ \begin{align} R'(X,Y)Z &= g(\nabla_X \operatorname{grad} f,Z)Y - g(\nabla_Y \operatorname{grad} f,Z)X\\ &+ g(X,Z)\nabla_Y \operatorname{grad} f - g(Y,Z)\nabla_X \operatorname{grad} f\\ &+ (Y f)(Z f)X - (X f)(Z f)Y\\ &- g(\operatorname{grad} f, \operatorname{grad} f)[g(Y,Z)X - g(X,Z)Y]\\ &+ [(X f)g(Y,Z)-(Y f)g(X,Z)]\operatorname{grad} f \end{align} $$

can this be used or adapted in case $f$ isn't $C^1$?

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First you have to understand what is curvature of singular metric.

I think the best possible answer in dimension 2 was given by Reshetnyak in and I suspect that the proofs work in higher dimensions. (Reshetnyak was interested in dimension 2 since in this case all reasonable metrics are conformally flat.)

Check "Two-dimensional manifolds of bounded curvature” by Reshetnyak.

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When you lose the regularity , the situation has to be evaluated case by case , I don't think there are " general procedures " to operate ...

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